About Slide Rules

No more than thirty years ago, slide rules were everywhere. Unlike many
instruments, instead of being steadily improved by the new electronic
technology, they were completely wiped out; sharing the fate of mechanical
cash registers and clockwork watches. It seems that few people today
know how to use a slide rule; most of my contemporaries who learned
at school have forgotten.

I had my first rule when I was about eleven, bought for me by my father to stop
me fiddling with his, and had
several
while at school and university. Like
everyone else, I bought an electronic calculator when they became affordable
around the early 1970s, and used it and its many successors for real work,
but I still retain my fascination for the simple old method, and I still
carry a slide rule and use it occasionally.

Put a slide rule on the table next to a modern scientific calculator. One
of them consists of half-a-dozen or so pieces of plastic with marks on, the
other contains the equivalent of millions of transistors, a few score of
electrical switches, a chemical power source, a display screen, and probably
at least as many pieces of plastic with marks on as the other.
Yet the calculator is faster, more accurate, easier to use and
cheaper. Amazing what technology can do, but also slide rules were always
far more expensive than they ought to have been. Though, of course, you
can make yourself a slide rule in a few minutes (see
here); try that with a calculator!

When slide rules were a reasonable means of performing calculations
quickly and cheaply, most science students were taught to use one at school.
Mostly, though, students seem to have been taught just the basic methods
for using the slide rule, and not the more efficient advanced methods, nor
much about how it all works, and therefore how to develop technique. In this
note I intend to set down what I've learned about it, before I forget it too!

Basics

The real basics:

A slide rule is a simple mechanical device to perform
multiplication and division.

How it works

The slide rule's operation is based on the mathematical property of logarithms, that:

log(a) + log(b) = log(a*b)

or in words that finding the product of two numbers is the same as finding the
number whose logarithm is the sum of the logarithms of the two numbers.

Here is a diagram of a basic slide rule, set up for multiplication by 2:

In the lower half of the picture is a scale running from "1" through "2" and "9"
back to "1" again. Consider the interval between the two "1" marks as a
unit interval; within this interval each number is plotted at a position
proportional to its logarithm. For example, "1", whose logarithm is 0, is
plotted right at the start of the interval; "2" (log 2 = 0.3) is
plotted just under a third of the way along; "3" (log 3 = 0.48)
is near the middle, and
"10", whose log is 1, is at the end of the interval. ("10" is written as "1"
for reasons which might become clear later.)

The upper scale is an identical
scale, but shifted to the right by 0.3 (log 2). With these two scales
positioned like this, it is possible to see how simple multiplication works.
Note that the "1" on the upper scale is aligned with "2" on the lower scale,
and also that "2" on the upper scale is aligned with "4" on the lower
scale, this is showing that 2 times 2 gives 4.
From the definitions of the scales, the distance from the left-hand "1" mark
to the "2" mark on both scales is log 2 = 0.3, and it can be seen that the
distances on the two scales add together to make 0.6, at which point on the
lower scale we find the number whose logarithm is 0.6, that is 4.

With the scales at the same position, any other number may be multiplied
by two, simply by finding the required number on the upper scale, and reading
down to the lower scale. For example 2*3=6 or 2*4.5=9.

What it looks like

This is a photo of a typical slide rule:

It consists of three parts, the stock, the slide and the cursor.

The stock is normally made either as two separate strips joined at the ends
by bridge pieces (as shown), or as a single piece with a recess cut out for
the slide to run in. In either case, the slide slides within the stock;
tongues on the slide engage with grooves in the stock to keep it in line.
The cursor slides along the stock, and has a clear window with a fine line
marked perpendicular to the slide motion. It is used to record positions
representing intermediate results, or to project from one scale to another
where they are not adjacent.
The rule shown above is double-sided, with a wrap-around cursor, but many
are single-sided.

The material is commonly high-quality plastic, but older rules tend to be
wooden, and expensive ones are often metal. Wooden ones are
faced with a thin layer of plastic to take the scale. Plastic scales are
engraved and filled with ink; metal rules may have a white background and
scale marks printed or silk screened on.

This is an old wooden Faber single-sided rule, fitted with two cursors:

It is conventional to put the scales that we have already seen, with one
unit interval for the whole length of the rule, at the lower sliding edge
on the front of the rule, that is the sliding scale is on the bottom edge
of the slide, and the fixed scale on the upper edge of the lower part of
the stock. These scales are conventionally known as "C" and "D"
respectively. Their position is shown on this image of the left-hand end
of the rule:

The diagram also shows two more scales, "A" and "B", at the upper slide edge.
These scales are plotted with a half-size unit interval, so
each scale runs from 1 to 100 in the same length that C and D run from
1 to 10. We'll come onto them, along with the dozens of other possible
scales, later.

Slide rules come in various sizes, and the size refers to the length of
the unit interval on the C and D scales. This length is typically 125mm
for pocket rules, 250mm for most rules or, rarely, 500mm. The two common sizes
are usually called 12-centimetre (or five-inch) and 25-centimetre
(or ten-inch) rules. Some rules, not necessarily American made, have the unit
length exact in old-fashioned units, for example 254mm for a "ten-inch"
rule. As we shall see later, the length of the rule is important as it
controls the maximum accuracy that can be achieved. The most accurate
slide rules abandon the linear slide and use helices to achieve scale
lengths of several metres.

Multiplication

We've already seen an example of multiplication above, but before going
on let's introduce some conventions to describe in words how to set the
rule. The form C:4 will be used to mean the position 4 on scale C. C:I
will mean the index of the C scale, which might be the 1 or the 10 mark.
In these terms, the process of multiplying 2 by 2 as shown above would be
written as: bring C:I over D:2; answer found on D opposite C:2.

We've seen how to multiply 2 by 2, but how about 20 by 200? The answer is
that it is done the exact same way. Except in certain special cases, slide
rule calculations are done ignoring powers of ten. Of course, the correct
power of ten is required in the answer, and usually this is supplied either
by knowing the approximate answer, or by working out the answer very
roughly on paper and using the slide rule just to supply more accuracy.
So, to multiply 20 by 200, we work out:

using the slide rule for the significant figures and mental arithmetic for
the powers of ten.

How do we multiply 3 by 5? Try the same procedure as before, and put C:1
over D:3. This puts C:5, where we look for the answer, off the end of the D
scale. In this case, we make use of the fact that we do not care what power
of ten we include in the answer. There's nothing under C:5, but if we
look exactly one unit length to the left of C:5, we'll find the result for
5 * 3 / 10, because subtracting one unit length is equivalent to dividing
by 10. This process is done in practice by using C:10 instead of C:1, which
moves the slide one unit to the left (let's call this "reversing the slide").
Bring C:10 over D:3, then read the answer, 1.5, under D:5.

There are rules which can be used to work out the power of ten for the
answer given the powers of the multiplicands and the exact way the slide
is moved, but in my opinion they are not much help except in simple cases.
For example, let's take 25 * 600. First reduce the arguments to the range
1..10 by extracting powers of ten, so:

25 * 600 =
2.5 * 6.0 * 10¹ * 10² =
2.5 * 6.0 * 10³

Then work out the figures on the slide rule. The power of ten for the answer
is the sum of the powers of the arguments, plus one for every time we use
the C:10 mark to start a multiplication. Here we have to use the C:10 mark
over D:2.5, to find 1.5 on D as the answer under C:6. So the power of
ten of the answer is 4, and the full result is 1.5 * 104 = 15000.
There
is a corrresponding rule for division (see below) where the power of ten
of the answer is decreased by one every time the result is found under C:10
instead of C:1. As I say, I think it's easier to approximate the result
overall than to keep track of the number of slide reversals.

Because the scales are logarithmic, the size of the divisions changes
as you go along the scale. Typically, a 25cm rule might have divisions
every 0.01 from 1 to 2, every 0.02 from 2 to 4, then 0.05 from 4 to 10.
This takes a little practice to read quickly. Also, to add to the confusion,
the labelling is often abbreviated. For example, the subdivisions between
1 and 2 are often labelled just 1 to 9, which really means 1.1 to 1.9.
In all cases, you are expected to estimate by eye down to about one-fifth of
a division.
Almost all scales have a special mark for pi, and often other constants
as well.

Division

Division is based on the formula:

log(a) - log(b) = log(a/b)

so uses subtraction of distances rather the adding we used for multiplication.
We can see this with the same setting of the rule as we used for multiplication
by 2, which also shows the position for, say, 6 divided by 3 gives 2.
You can see how the distance 1..3 on C, which is log 3 = 0.48, is
subtracted from the distance 1..6 on D, which is log 6 = 0.78, to leave
the distance 1..2 on D, which is 0.3 = log 2

The basic rule is: align the divisor (number to be divided by) on C with the
dividend (number to be divided) on D, and read the answer under the index
of C. For example, slide C:3 over D:6 and read the answer, 2, under C:1.

As with multiplication, for some pairs of numbers, C:1 will be off the end
of the D scale; then the answer will appear under C:10. This will actually
be ten times the answer, as we're reading one unit interval to the right,
but as usual we are not concerned with powers of ten.

Another example: 365/12. Set C:1.2 over D:3.65 and read the answer 30.4
under C:1. In this case, as we're working out the average number of days
in a month, we can guess the powers of ten!

When aligning two numbers, neither of which are on an exact division,
it may be helpful to place the cursor on the dividend on D, then bring the
divisor on C under the cursor line. Where one or more of the arguments
are on exact divisions it may be more accurate not to use the cursor, but
even then it can be helpful to put it nearby to keep the eye focused on
the right part of the scale.

Chained calculations

By a chained calculation, I mean something like a*b*c, or a*b/c.

When
there are repeated multiplications, the cursor is used to hold the result
of one calculation on D, while the index on C is brought up for the next.
For example, let's do 2*3*4*5. Place C:1 over D:2, then bring the
cursor up to C:3, thus marking the first intermediate result (2*3=6). Then
bring C:10 under the cursor line and move the cursor to C:4, marking the
second intermediate (2*3*4=24). Then bring C:10 again under the cursor line,
and read the final result under C:5 (2*3*4*5=120). (The powers of ten rule
would say two reversals of the slide, so add two powers, so result =
1.2*10²
= 120.) Note that each multiplication involves one slide movement and one
cursor movement.

Repeated divisions work much the same way. Let's try 1/(2*3*4). First
align D:1 with C:2, then place the cursor at C:10, marking first intermediate
(1/2 = 0.5) on D. Then bring C:3 under the cursor and move the cursor to C:1,
which is the second intermediate (1/6 = 0.167). Finally bring C:4 under
the cursor and read the result 0.0417 under C:10. (The rule would say
C:10 used for result twice, so deduct two powers, so result =
4.17*10-2 =
0.0417.) Note again that each step requires a slide and a cursor movement.

Alternating multiplication and division is the most efficient sequence when
only C and D scales are available. To see this, let's try (4*5)/(8*0.125).
Calculated in the order in which it is written, it would go: C:10 over D:4,
cursor to C:5, C:8 under cursor, cursor to C:10; C:1.25 under cursor, result
D:2 under C:1. That's 6 movements, including reading the result. Alternating
it would go: C:8 over D:4, cursor to C:5, C:1.25 under cursor, result under
C:1 = D:2. That's four movements, including reading the result.

Working this through, you can see why it's more efficient to alternate, as
division leaves the intermediate result under C:I, which is where it is
needed for multiplication, so there's no need to mark it with the cursor.
Similarly, multiplication leaves the result marked with the cursor on D,
which is where it is needed for the next division. Of course, it might work
out that a division leaves the slide set so that the next multiplier on C is
off the end of the D scale, in which case the cursor has to be moved and the
slide reversed, reducing the saving. Other scales, such as CF and DF help
to reduce this. Also, other techniques using reciprocal scales can make
repeated multiplication or division as efficient as alternating.

Square scales (A, B)

Before starting with more scales, a note about names. I haven't seen a rule
yet that doesn't have C and D scales on the lower sliding edge on the front.
All but one of mine has A and B scales at the upper sliding edge on the
front. These scales are almost universally labelled by the letters A, B, C and D
on the left-hand end of each scale. Beyond these, any combination of scales
may be found, and most of the other scales have names which either denote
their function, such as S for sine, or their relationship to C or D, such
as CI for C inverse. These names for these scales also seem to be pretty
universal. The old wooden rule in the picture above has only ABCD scales
on the face, in the conventional place but unlabelled, though the S, T and
L scales on the back of the slide are labelled.

In addition to the names on the left of each scale, most rules have the
derivation of each scale, from the base CD scales, shown by a formula at
the right-hand end. In these fomulae, the C and D scales are represented
by x, so they both have x at their right-hand ends. Scales A and B, with
half the unit interval, are both labelled x² (x-squared). To see why
this is so, think about the mid-point of the A-scale, which represents
10. This point is vertically above the mid-point of the D-scale, say
at some reading x. Now as this is the mid-point, log(x) = 0.5, or x=sqrt(10).
Reading backwards, sqrt(10) on D reads onto A as 10. This is true for
any position on D, so any number on D reads onto A as its square. Thinking
about it another way, the factor of two scale change in linear distance
terms between A and D represents a squaring by the equation:

log(a) * 2 = log(a²).

Apart from this use of calculating squares and square roots, the A and B
scales can be used for multiplication and division in the same way as
C and D, but of course with reduced accuracy, though this is compensated
by the reduced need to reverse the slide. For example, to do our previous
example of 2*3*4*5 on A and B scales we could do: B:1 over A:2; cursor to
B:3; B:1 under cursor; cursor to B:4; B:100 under cursor; answer under B:5
= A:120.

In finding square roots by reading from A to D, it is important to think
carefully about powers of 10, as a difference of one power on the A scale
translates to sqrt(10) on D, which is not just a matter of adding a zero!
The rule is to reduce the number to a value between 1..100 and an even
power of ten, take the root and then put back half the power of ten. Some
examples:

Reading from A to D is of course done with the cursor. In calculations
that require a square or a square root, it is often possible to get clever
and do part of the calculation on AB and part on CD, crossing when the
second-power operation is called for. For example, to calculate 8 *
(.2*16)²
proceed thus: C:1 over D:2; cursor to C:1.6; B:1 under cursor, cursor to
B:8 and read the result at A:82. Of course, great care is needed if a
square root is to be taken, to ensure the required even power of ten
exists.

Cube scales (K)

There is usually only one K scale, on the stock, marked x³ (x-cubed),
with three logarithmic cycles in the space of one on the C scale,
running from 1 to 1000.
Reading from D to K gives the cube, and from K to D the cube root. As
with square roots, it is necessary to get the powers of ten under control.
With cube roots, the number must be between 1..1000, and the power of ten
a multiple of 3. An example:

I have one odd rule that has two K scales (K and K') at the upper sliding
edge on the reverse, so that calculation can be carried on after cubing.
Given the limited accuracy of these scales, I don't see why anyone would
want to.

Reciprocal scales (CI, DI)

CI is very often present; DI is unusual. The scales are marked 10/x or
sometimes 10:x at the right-hand end. They are plotted with the same unit
interval as C, but the marks are plotted at 1-log(a) instead of at log(a)
for C, so they look the same as C or D, but run backwards from 10 to 1.
CI is on the slide, and DI is on the stock. Their formula is:

1-log(x) = log(10)-log(x) = log(10/x)

and reading from C to CI or from CI to C produces the reciprocal, ignoring
powers of ten.

The main use of CI especially is not to find reciprocals per se, but for
speeding up chained calculations. Remember that multiplication by x is
equivalent to division by 1/x, and you can see that is it possible to
turn a multiplication operation, where C:I is placed, into a division-like
operation, where the operand is placed, if that is more convenient in a
chain. Let's run that 2*3*4*5 again, using this technique: put the cursor
on D:2; put CI:3 under the cursor; move cursor to C:4; put CI:5 under
cursor; read result under C:1 (=120). This is only five movements, including
reading the result, as opposed to six the other way, and there was no
need to think about reversing the slide.

As you work through a calculation, at each stage you can make a tactical
decision whether to proceed by the obvious multiplication or division, or
whether to use the CI scale and the opposite operation, based on convenience,
and positioning for the next operation.

I've seen no use for DI.

Pi-shifted scales (CF, DF, CFI, DFI)

These scales are marked pi-x, and when present are often at the upper
sliding edge on the reverse. They are normal unit-length scales, but
shifted to the right by log(pi), so that crossing from say D to DF gives
a multiplication by pi, and from DF to D gives a division by pi. This
can be useful in itself, but the prime purpose of these scales is to
facilitate chained calculations and tables. As pi is close to sqrt(10), these
scales are shifted by about half a basic unit; this means that until the slide
is more than half-way out, there is always at least one incidence of each
value on C or CF that is aligned with D or DF, and by carrying on the
calculation on the appropriate pair slide reversals can be eliminated.

This is particularly obvious when setting up a table. Say that we want to
convert some prices from pounds to marks, with a conversion factor of
*2.7. We set the rule so that C:1 is over D:2.7, then by moving the cursor
along, any price in pounds on C can be converted to marks on D, so 3 pounds
converts to 8.1 marks. However, pound amounts from about 3.8 to 10 on
the C scale are
off the end of the D scale, so normally we would have to keep reversing the
slide, to put C:10 over D:2.7, depending on the amount to be converted.
But, without moving the slide, we see that on CF amounts from about 3.2 to
11 pounds are over DF, so we can convert 7 pounds on CF to 18.9 marks on
DF.

When working with CF and DF, it's important to keep in mind whether the
cursor is set to a value on D or on DF. When you're looking for a value on
C, you can just as well use CF, but once you've chosen which and placed
the cursor, you must use the corresponding scale for the next operation.
If you've read from CF to a value on DF, saving that DF value with the
cursor, you must place the next division operator on CF under the cursor,
not C, or if the next operation is a multiplication, you must use the
CF scale's index. If this sounds complicated, it's probably because it
is. Few people who don't use a slide rule all the time would bother
with CF and DF except for tables.

Logarithm scales (L)

The logarithm scale looks different from all other scales because it is
a linear scale, like a ruler. It is labelled log x. It is usually
marked 0.0 to 1.0 along the unit interval, so reading from C to L gives
the fractional part of the logarithm, from the definition of the C scale.
The integer part of the logarithm must be supplied, as the power of ten.
For example, log 400: C:4 lies over L:0.602; 400 is 4 * 10²; so log 400
is 2.602. Similarly, reading the other way gives 10x, so for
102.3: 102.3 = 100 *100.3; L:0.3 corresponds
to C:1.995; so 102.3 is 199.5.

If a rule had two L scales, one on the slide and one on the stock, it would
be possible to do simple addition, but pretty pointless.

Trigonometrical scales (S, T, ST, T2, P)

These scales are basically just look-up tables for common trigonometric
functions. They are usually marked in degrees, or in some cases degrees
and minutes. Their range is fairly restricted. These scales are the first
that we have met that cannot do arithmetic on their own, they are just
tables, and what's more they are not invariant under factors of ten. Each
scale relates to just one value on the corresponding base scale, so to
cover a wider range of angles, there have to be more of them.

The commonest set is probably ST, S and T, defined as arc(0.01x) (ST),
arcsin(0.1x) (S) and arctan(0.1x) (T). These scales would usually be on
the stock, and x in the formulae is the value of the D scale. Arc is
used to stand for arcsine or arctangent when the angle is so small
that they are indistinguishable
at the resolution of a slide rule. The ST scale shows the angles whose
sines are in the range 0.01 (opposite D:1) to 0.1 (opposite D:10), roughly
0.6 degrees to 6 degrees. The S scale covers 6 degrees to 90 degrees (sines
in the range 0.1 to 1) and the T scales cover about 6 degrees to 45 degrees
(tangents in the range 0.1 to 1). Sometimes T is called T1, and an extended
tangent scale, T2, is supplied for angles from 45 to about 84 degrees; this
has formula arctan(x). Often the scales are also marked with red numbers
running the opposite way, to remind the user that cos(x) = sin(90-x) and
cot(x) = tan(90-x).

The scales are mostly used for looking up trig functions, but they can
also be incorporated into calculations; to work out 5*sin(25) put C:I
above S:25 (using the cursor); read the result on D opposite C:5 (=2.115).
The blooper to avoid is to try calculations using the angles: the trig scales
are not themselves logarithmic, you cannot calculate 2*arcsin(.5)
directly. You must first calculate arcsin(.5) by cursor to D:5, read
S:30; manually transfer this to D:3 (for 30); then multiply by 2 in the
usual way on C and D.

Another common pattern is to have just two trig scales, on the back of the
slide (of a single-sided rule); they are brought into use by pulling the
slide right out and flipping it over. In this case, the upper scale,
usually S, is marked with reference to the B scale which it temporarily
replaces, and the lower scale, T, is marked with reference to the C scale
which it replaces. So the T scale is the same as before, but the S scale
runs from 0.6 degrees to 90 degrees, with 6 degrees in the middle, and
the left-hand half of it also acts as an ST scale for tangents of small
angles. This
arrangement is sometimes convenient for calculations, removing the need
to carry across with the cursor, but loses accuracy on S and is generally
only found on low-spec rules. These rules often have small windows on
the back which allow sine and tangent look up to be done without flipping
the slide over; you line up your angle on say the S scale with the mark
in its window, then read the corresponding sine on the front under A:100.

For angles less than 0.6 degrees, the sine and tangent are so close to
the angle in radians that conversion is done by multiplication by pi/180.
There is often a mark at C:1.745 or its reciprocal C:5.73 to make this
easier. Sometimes there are other marks for angles in minutes and seconds.

Another possibility is the P scale, for Pythagoras, which is scaled as
sqrt(1-(0.1x)²). This can be used to convert the sine of an angle to
its cosine without going through the angle (in the range 0.1 to 1 only),
but I've never had cause to use it.

Differential Trigonometrical scales (SD, TD, ISD, ITD)

Some Thornton rules have idiosyncratic trig scales, which instead of
directly looking up the trig functions of an angle instead look up the
ratio between those functions and the angle itself. These scales are very
short, which allows sine, tangent and both inverses to fit into the
length of one conventional scale. Their formula is difficult to write
in the conventional format where the scale value is given as a function
of x, the other way round giving the value of x at which the angle T is
plotted they are x = T/sinT for SD and x = T/tanT for TD. To find the sine
of T: set T on SD over T on D and read result on D under C:I. The inverse
trig scales read to CI scale so that a division operation actually performs
a multiplication. To find the angle whose tangent is x: set x on D aligned
with x on ITD and read the result on D under C:I. These scales give a
better range of angles, effectively automatically using sin(x) ~= x for
small angles, and are claimed to give better accuracy everywhere, but like
most clever tricks few users will remember how to use them from one time
to the next.

Log-log scales (LL1..., LL01...)

The log-log scales are another set that are not power-of-ten invariant, so
again there is a whole family of them, the posher the rule, the more of them
there are, though I've not seen more than eight on one rule. The basic
formula of LL0 is e0.001x, of LL1 e0.01x, of LL2
e0.1x and of LL3 ex. The basic formula of LL00 is
e-0.001x, of LL01 e-0.01x, of LL02 e-0.1x
and of LL03 e-x. The commonest set is just LL2 and LL3, and in
general the positive exponent ones are commoner than the negative.

The scales are used for taking powers and roots of numbers, and for
taking logarithms to other bases. They have the disadvantage that, except
for a limited range of numbers, they are horribly approximate, as the
scales are very compressed for the larger values of x.

The LL scales can be used as straight look up tables, to find natural
logarithms and powers of e. For example, to find ln(pi): use the cursor to
project LL3:3.14 to D (=1.14).

Unlike the trig scales, it does make sense to do arithmetic with the
values on the log-log scales. The numbers on the LL scales are plotted at
their logarithmic position, onto a logarithmic scale, so using the single-log
C-scale alongside the double-log LL scales is equivalent to multiplying the
logs of the numbers on the LL scales, which means exponentiation.
The formula is:

log(ln(x)) + log(a) = log(a*ln(x)) = log(ln(xa))

(where 'x' is the reading on the LL scale and 'a' the reading on C)
so to find 32.5: set C:1 over LL3:3 and read the result (=15.6)
on LL3 opposite C:2.5. To find 29: set cursor on LL2:2; set C:10
under cursor; read
result on LL3 under C:9 (=~500). Note how, because using C:10 instead
of C:1 loses a power of ten, we had to read the result from the next
higher log-log scale to get the power of ten back again.

Another calculation possible on the LL scales is to take log to any base.
Remembering that, say log2(6) is the power that we have to raise
2 to to get 6, and that raising to a power is like multiplying the LL scale
by the C scale, place C:10 over LL2:2, and then place the cursor over
LL3:6; we find the required result on C at 2.58. Again we used C:10
rather than C:1, but compensated by reading from LL2 to LL3. Another
example: log5(20) = 1.86 can be done using the values on C
directly, and entirely on the LL3 scale, and another: log2(1.2)
= 0.263 can be done entirely on LL2, but use of C:10 rather than C:1 makes
the result one tenth of the C scale reading.

The LL0x series scales do the same thing, only for numbers less than one.
If they're not there, you can often use the LLx scales instead, by
taking the reciprocal of the numbers. So to find e-2 we can read
from D:2
to LL03 to find 0.135, or read to LL3 at 7.4 and return the result 1/7.4
=0.135.

Differential log-log scales

These are a feature of some Thornton slide rules, and effectively provide
an LL1 scale with just a very short extra scale at the end of the rule,
to the left of the A scale. The LL1 scale is very useful for interest rate
calculations as it covers ratios from 1.01 to 1.11, or 1% to 11%. The
markings of LL1 are quite close to D scale markings in that 1.01 is just
left of D:1, 1.04 a little further left of D:4 and so on. The little
differential scale just provides a convenient way of offsetting this small
displacement. Setting the extra "Y" cursor against 4% on the differential
scale and reading D against C:4 reads the equivalent result to setting the
cursor to LL1:1.04 and projecting to D.

At the next level down, for the LL0 scale from 1.001 to 1.01, the marks
are so close to the D scale that they can be used equivalently; some Faber
rules use this to share the D scale with LL0.

Accuracy and double-length scales (W1, W1', W2, W2')

These are pretty esoteric. Their function is to provide increased
accuracy for multiplication and division, at the expense of considerable
extra complexity. Typically, they occupy both sliding edges on one side,
stretching a single 1..10 unit range into the total length of both.
The lower scales are called W1 (where D would be) and W1' (instead of C);
the upper set are W2 (instead of A) and W2' (instead of B). Their
formulae are sqrt(x) for W1 and W1'; sqrt(10x) for W2 and W2'.

A diversion on the subject of accuracy. We have seen that logarithmic
scales have the property that any given linear distance corresponds on the
scale to a fixed multiplier, that's how multiplication works. So any
error in reading, which is likely to be the same physical distance wherever
we are on the scale, corresponds to a constant ratio error in the reading,
and a constant percentage error. This means that the accuracy of a slide rule,
depending as it does mainly on the human eye reading and estimating between
engraved marks, is better the longer the unit cycle. Suppose we take the
reading error to be 0.1mm, which would be pretty good, then on a 250mm
rule this represents a ratio error of 100.1/250 = 1.00092 (you can
do that calculation on a slide rule if you have LL3 and LL0 scales!) or about
0.1%. For a 125mm it's 0.2%, and for a 500mm it's 0.05%. This factor is
applied for each slide or cursor operation, so for 6 operations expect
no better than 0.6% on a 250mm. Greater accuracy can be got only by longer
scales or finer divisions, and presumably a microscope to see them with.

So, a cheap way to double your accuracy is to double the scale length,
but given the inconvenience of carrying or using the rule this cannot go
very far. Hence the compromise of double-length folded scales.

W scales are not too bad if you keep your wits about you. The lower W1
scales start at 1 and stop at a red mark where 10 on D would have been,
we'll call this W1:R. The upper W2 scales start at a red mark called
W2:R and run up to 10. The red marks are important, as they get used
in place of the normal scale indexes. Think of them as marking sqrt(10),
so when they are used they inject spurious factors of sqrt(10). However,
as the W2 scales are offset by sqrt(10) from W1 (they're offset one
half of a unit length, remember), the spurious sqrt(10) factors can be
taken out again by crossing the rule. So we get a basic principal: if
your calculation used a normal index, read on the same side; if it used
a red mark, cross the rule for the answer.

Let's try an example: 2*2. If you set W1':1 over W1:2, W1':2 is off the
end. So, set W1':R over W1:2 instead, then move the cursor to W1':2. Now,
because we used a red mark, don't read the result on W1, but cross the
rule and read it on W2, to find the result 4. It works!

You don't have to use the red marks, you can use the proper index marks
on the opposite side and line up using the cursor, then it's even more
obvious that as you crossed over to set the index you should cross back
to read the result. I think the red lines are easier.

Let's try a division, say 2/6. 2 is on W1 and 6 is on W2, so set the
cursor to W1:2 and bring W2':6 under the line; read the result 0.333 on
W2 opposite W2':R. We use the red mark as we crossed sides when we set
up the division.

That's about it, as there are no short-cut facilities, no equivalent
of CF and DF or CI; also no trig functions or other fancy scales based
on the W scales. You can do basic multiplication and division more
accurately, that's all.

Square and cube root scales

I've only seen these on an American Pickett rule. They are effectively
double and triple length scales, but on the stock only so they can't be
used for high-accuracy multiplication and division. The do provide more
accurate squaring, square rooting, cubing and cube rooting than a normal
rule.

Other scales

The only scales not yet mentioned that I have on any of my rules are
dedicated electrical ones. These operate with special marks on the ABCD
scales to calculate voltage drop in copper cables and the efficiency
of an electric motor or generator. They are so specialised that without
the instructions it's easier to do the calculations on the general scales,
unless it's something you do every day.

Cursor lines

Some makers of slide rules pack the cursor with extra lines. As the fixed
spacing between two lines on the same scale corresponds to a multiplication
factor, simple conversions can be arranged, provided the factor is close
enough to one to fit onto the cursor. A common one is kilowatts to horsepower,
at 735 watts or 746 watts depending on country of origin.

Another, almost universal, one is to read from 2 on D to pi on A; this one
combines two factors and a squaring to calculate pi*(x/2)², or the area
of a circle of diameter x.